The tree packing conjecture for trees of almost linear maximum degree

TL;DR

This paper proves that a large collection of trees with almost linear maximum degree can be packed into a complete graph or a quasirandom graph, extending to certain bounded degeneracy graphs with specific structural properties.

Contribution

It establishes a packing result for trees with nearly linear maximum degree into complete and quasirandom graphs, generalizing previous results and allowing broader graph classes.

Findings

Trees with maximum degree up to $cn/\log n$ pack into $K_n$

The packing extends to quasirandom host graphs

Results apply to graphs of bounded degeneracy with specific structural features

Abstract

We prove that there is $c>0$ such that for all sufficiently large $n$, if $T_1,\dots,T_n$ are any trees such that $T_i$ has $i$ vertices and maximum degree at most $cn/\log n$, then $\{T_1,\dots,T_n\}$ packs into $K_n$. Our main result actually allows to replace the host graph $K_n$ by an arbitrary quasirandom graph, and to generalize from trees to graphs of bounded degeneracy that are rich in bare paths, contain some odd degree vertices, and only satisfy much less stringent restrictions on their number of vertices.